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Game Theory Calls Cooperation Into Question
An anonymous reader sends this excerpt from Quanta Magazine:
The physicist Freeman Dyson and the computer scientist William Press, both highly accomplished in their fields, have found a new solution to a famous, decades-old game theory scenario called the prisoner's dilemma, in which players must decide whether to cheat or cooperate with a partner. The prisoner's dilemma has long been used to help explain how cooperation might endure in nature. After all, natural selection is ruled by the survival of the fittest, so one might expect that selfish strategies benefiting the individual would be most likely to persist. But careful study of the prisoner's dilemma revealed that organisms could act entirely in their own self-interest and still create a cooperative community.
Press and Dyson's new solution to the problem, however, threw that rosy perspective into question (abstract). It suggested the best strategies were selfish ones that led to extortion, not cooperation.
[Theoretical biologist Joshua] Plotkin found the duo's math remarkable in its elegance. But the outcome troubled him. Nature includes numerous examples of cooperative behavior. For example, vampire bats donate some of their blood meal to community members that fail to find prey. Some species of birds and social insects routinely help raise another's brood. Even bacteria can cooperate, sticking to each other so that some may survive poison. If extortion reigns, what drives these and other acts of selflessness?"
Press and Dyson's new solution to the problem, however, threw that rosy perspective into question (abstract). It suggested the best strategies were selfish ones that led to extortion, not cooperation.
[Theoretical biologist Joshua] Plotkin found the duo's math remarkable in its elegance. But the outcome troubled him. Nature includes numerous examples of cooperative behavior. For example, vampire bats donate some of their blood meal to community members that fail to find prey. Some species of birds and social insects routinely help raise another's brood. Even bacteria can cooperate, sticking to each other so that some may survive poison. If extortion reigns, what drives these and other acts of selflessness?"
Why isn't this headline, "Game Theory Called Into Question for Failing to Predict Observed Examples of Cooperation?"
Real Life isn't Spherical Cows. They need a better model.
Sometimes the 'fittest' thing to do is to help your community because having a strong community helps you more than going it alone.
Iterated Prisoner's Dilemma contains strategies that dominate any evolutionary opponent
The selfish gene theory popularized by Richard Dawkins states that evolution works on genes, not on individuals. Any gene which gives rise to behavior that will cause more copies of that gene to survive, will increase its percentage in the gene pool at large.
http://en.wikipedia.org/wiki/G...
Nature includes numerous examples of cooperative behavior...If extortion reigns, what drives these and other acts of selflessness?"
Any individual benefits of altruism aside, the potential for cooperation improving the fitness of a species is clear.
.: Semper Absurda
News at 11
Politics leads to some extremely sub-optimal solutions, thus the expression about the Perfect being the enemy of the Good. Extortion is a good solution.
[Theoretical biologist Joshua] Plotkin found the duo's math remarkable in its elegance. But the outcome troubled him. Nature includes numerous examples of cooperative behavior. For example, vampire bats donate some of their blood meal to community members that fail to find prey. Some species of birds and social insects routinely help raise another's brood. Even bacteria can cooperate, sticking to each other so that some may survive poison. If extortion reigns, what drives these and other acts of selflessness?"
I'm not sure Joshua Plotkin read the paper. It does not claim (as I understand it) to represent every scenario, merely a special case of a specific scenario. Explicitly, it requires the organism to have enough intelligence to remember what happened in previous games, so a bacteria without memory is not covered under this model. The strategy requires multiple rounds be played.
Also worth mentioning that 'good for the individual' is not the same as 'good for the species,' and nature selects the latter
I know almost nothing about vampire bats (except don't get bit, you'll need rabies shots!), but if someone understands how it relates to the prisoners' dilemma, I'd be interested in hearing it.
"First they came for the slanderers and i said nothing."
In other news, some people believe economic theory.
It's that simple. They have a neat mathematical model which is interesting, but if it doesn't make accurate predictions when applied to a more realistic scenario then it's missing something.
BeauHD. Worst editor since kdawson.
"Well, I don't know if I believe anyone is 100% a dick..." Rhomann Dey
I take my children to see Madonna(..), but I never for once ever thought I was in the same business.Chris Rea.
... The age old, racking up favors-owed.
Did Dyson extort Press, or did Press extort Dyson?
It's about procreation and the survival of the genetic line. Individual survival is irrelevant, especially once one has procreated. (Though even those who don't contribute to the survival of the genetic line of their family - the person who has a sibling willing to sacrifice themselves to save the family enhances the chances the family will procreate.)
This kind of confusion is what happens when people try to do research outside of their expertise. If you want to understand biology, ask a biologist, not a physicist or a computer geek. (Though a lot of biologists make the same mistake, of course.)
WTF. The examples with cooperating organisms are irrelevant since evolutionary pressure acts on genes foremost, and not individuals. The question is whether the outcome with extortion would be as worrying if you apply it to genes instead.
The physicist Freeman Dyson and the computer scientist William Press [..] have found a new solution to [...] the prisoner's dilemma
So... what you're saying is... these two guys have cooperated to call cooperation into question...
Riiight...
systemd is Roko's Basilisk.
If extortion reigns, what drives these and other acts of selflessness?"
All the cooperating ones use Richard Stallmans GPL
Darwin never suggested "survival of the fittest". What does this even mean? 'Fittest' must mean ' most fit in a certain environment', but how is that measured? 'Most fit' must can only be meaured as the ones 'that survive'. So the statement can only mean "survival of the survivers" which is a trivial obsurdity.
don't cut it off www.mgmbill.org
I wonder if this has any relation to the role of 'spite', which is sort of altruism's barbed cousin: (for my purposes, 'spite' in the sense of 'inflicting injury on someone else without benefit, or even at direct cost, to oneself')?
Spite is hardly a directly rational response, especially if you are inflicting injury as a reprisal for something that somebody has done to a third party, rather than to you; but it is clearly something that humans do(or, even if they don't, they often fume indignantly and wish that they could); and seems that it could provide additional incentive to behave altruistically, or at least within the bounds of 'fairness'
The researchers were going to publish the study, but they wouldn't cooperate with the publishers.
Table-ized A.I.
Bats helping other members of the herd stay healthy -- there's probably some benefit to being surrounded by a healthy herd. Improved access to healthy mates, more efficient hunting, security in numbers.
Fittest means most well adapted to the environment it finds itself in. That means cooperation is often the best solution.
Also, prisoner's dilemma is a bit like spherical cows. In the wonderfully complex and varied real world, beggaring your neighbour might get the best instaneous result right now, but tomorrow when you need your neighbour's help, you're screwed...
Their model isn't necessarily inadequate. Perhaps the cooperative strategy was simply easier to arrive at through evolution. The extortion strategy might be in a hard-to-reach part of behavioural state space. It's taken these brilliant mathematicians a good while to find it, after all. If evolution finds a suboptimal, but still beneficial strategy, it can be hard to subsequently jump out of that local minima to reach an even better solution.
You can download "On the origin of species" by Charles Darwin for free from many different places.
That sucker was published in 1859 and not even Mickey Mouse can keep it out of the public domain.
So they found out that sometimes populations get stuck in situations without cooperation and that this is not beneficial for their payoff and for the population as a whole. So how is this showing cooperation is bad or impossible? It just shows that sometimes you get stuck and there is no way out. How does that calls cooperation into question? Only in /.tards eyes maybe?
"If man were not selfish, the world would be destroyed."
Please stop putting a negative spin on selfishness. There is no such thing as unconditional altruism (contrary to what various religious doctrines would have you believe). There's a condition, a price attached to every word, deed, thought.
When the manuscript crossed his desk, Joshua Plotkin, a theoretical biologist at the University of Pennsylvania, was immediately intrigued.
Remind me again why I care about some dude's reaction.
If you look at the abstract, this was published in 2012.
Actually we see this in the history: non-cooperative societies, built on violence and slavery gave way to much better societies, despite single people stil being egoistic.
I actually read this book. Apparently you did not. The term 'survival of the fittest' is a misunderstanding of evolution as well as misquoting Darwin.
Herbert Spencer coined the phrase "survival of the fittest". Darwin used the term "natural selection"
don't cut it off www.mgmbill.org
Altruism is an evil that must be stamped out. Mercy for the weak only oppresses the full potential of the human race. Take that statists!
He's right though - the phrase does not exist anywhere in any of Darwin's works.
Darwin hired a journalist by the name of Herbert Spencer and tasked him with explaining his theory to the public at large. While his book sold incredibly well (breaking several bestseller records -quite remarkable for a science book) Darwin was concerned it would not be well understood by people who lacked a science background. Spencer's job was to explain his theory of natural select - what we now call evolution - another term he never used to the public in layman's terms.
This had the major advantage that most people now thought they understood Darwin's theory.
It had the major disadvantage that most people now thought they understood Darwin's theory.
Anyway, it was Spencer who coined the phrase "survival of the fittest" to explain that aspect of the theory, Darwin never said it or anything like it.
Unicode killed the ASCII-art *
What really is the difference?
If you want your slaves/servants/employees to generate value for you continuously, you have to give them the illusion of giving selflessly.
Survival of the fittest for the individual. Survival of the fittest for the species.
The last two squirrels in the tree, one single nut. The selfish squirrel eats, the other starves. Selfish squirrel doesn't get to breed, tree becomes free of squirrels.
There's probably a point during a co-operative species evolution at a given geographical locale where the factors of population density, predators, resources, brood size etc results in negative growth without cooperation.
Meanwhile in the next tree, there's lots of nuts, all the squirrels are selfish. A few years later there's a squirrel orgy and both traits are mixed into the next generation.
Just like heat shock proteins - it's not hot now but it clearly was at some point. So the population density of a cooperative species might seem fine now but perhaps it was once squeezed and gave rise to cooperation traits in that species and we now can't see the reason because the pressure that gave rise to the behaviour is no longer present.
Or maybe Richard Dawkins is responsible somehow.
"Prisoner's dilemma" is actually rather a broad term. Do you mean infinitely repeated prisoner's dilemma? (Solution: Both cooperate.) Or do you mean one shot prisoner's dilemma? (Both defect.) Or to N-iterations when N is unknown? (Solution depends upon the probability of each turn being the last.) Or perfect information to N-iterations where N is known? (Solution: They defect from the get go.) Or imperfect information to N-iterations where N is known, but neither side is sure if the other side is rational or a good person? (Solution: Even with two perfectly rational self interested parties, they cooperate for a while then defect towards the end - lame duck president anyone?)
Prisoner's dilemma is a base line to work from and see how stuff changes. Not the be all and end all of game theory.
And no, prisoner pairs are not going to compete. If one of the prisoners from the first pair had decided to cooperate with the other, that wouldn't change that the first prisoner defects, so he just gets doubly screwed. If each players' move is not independent of the others for that iteration, then you aren't talking about prisoner's dilemma.
> "... If extortion reigns, what drives these and other acts of selflessness?"
Egotistical individuals are of the same species (in Nature, e.g. like bats) -- hence, they are not perfectly independent. One losing means the other loses something.
The question now becomes: why does symbiosis arise between different species?
Maybe they might still not be independent -- this time because of consumption habits and necessities.
PS: Interestingly, this also applies to humans, but on so many more levels. For instance, losing a tribe might mean losing ancient knowledge which prevents cancer. Or a biological trait which works to the same end (I recall reading about dwarves -- in Chile? Ecuador? -- which exhibit that same characteristic). Also "the enemy", once defeated, often means a lot of land was devastated by bombs and some mankind heritage destroyed beyond recovery.
Producing more offspring that manages to procreate.
'Most fit' must can only be meaured as the ones 'that survive'.
No, survival is a necessary, but not a sufficient criterion. Reproduction rate is what actually counts. If there are two groups with different reproduction rates, the larger one will eventually become completely dominant, especially when the two groups start getting into conflicts about resources.
The paper seems to be showing that if you assume that your opponent's strategy is fixed, you are subject to extortion. The authors then say that not assuming that your opponent's strategy is fixed means having a theory of mind, and that requires sentience. The argument doesn't work because you don't need a "theory of mind" or "sentience" in order to model strategy switching. A pretty poor paper.
It's hardly new, as the paper was published over 2.5 years ago...
..."Scientists Still Trying To Determine Who Exactly Was On First".
It's an imaginary problem involving perfectly rational actors. Humans are NOT rational. End of story.
Trying to "solve it" with humans is like trying to calculate a joke.
Mit der Dummheit kämpfen Götter selbst vergebens
You have to look around to understand that our world is dominated by such a co-operation. But as it happens in nature, you can recover your intuition with sensistivity analysis. Which of the two choices is more robust under noise?
Evolution can only occur in a non--zero-sum environment -- energy.in > energy.out for an organism or collection of organisms, if only by a hair, otherwise you get stasis, and ultimately death, when a more dynamic organism begins to farm you. Cooperation is a strong strategy, and Dawkins' insight that selection operating at the gene level can account for altruistic behavior (altruism being the ultimate cooperative behavior) at more macro scales was nothing short of brilliant. This paper pretty much provides a model that accounts for one player co-opting the successful strategies of other players in the iterated prisoner's dilemma. If you play an MMORPG, you are participating in this model of co-opted cooperative behavior. Take WoW, for example: inferior players are paying Blizz for the privilege of being farmed by superior players for gold, or PvP honor points.
All this paper shows is, once tit-for-tat establishes a beachhead and converts the population to a largely cooperating one, other strategies will emerge that will exploit the naive cooperators. Big deal. It is very well known.
Within two generations of eradicating most viral diseases, the anti-vaccination people are back, showing that once something becomes the common wisdom, there will be incentives to be a contrarian.
sed -e 's/Chuck Norris/Rajnikant/g' joke > fact
Who's to say the international market isn't extortion?
Is printing money extortion?
Is the money system extortion?
How would society function without being backed up by violent police?
How would international diplomacy go without being backed up by military, nukes and threats of total anihilation?
The faster we own up to this, the better.
Extortion sounds about right...
Now, on an individual and local level, that's another matter entirely!
Using the prisoner's dillemma doesn't account for the other prisoner's shanking you if you get to be too big of a prick
The shepherds did so well protecting the flock that the sheep no longer believed that wolves existed.
He's right though - the phrase does not exist anywhere in any of Darwin's works.
Not true.
http://darwin-online.org.uk/content/frameset?pageseq=1&itemID=F878.2&viewtype=text
Bottom of page 89, Animals and plants under Domestication, Charles Darwin.
"...and from the survival of the fittest, ..."
One of many examples.
Darwin used both. He wrote more than one book.
I declare the use of online searchable databases of facts relating to the argument at hand unfair competition in online discussions!
So I read through the paper, and it was certainly above my maths, but it seems the most important point was actually left out. If I understood it correctly the "extortionate" idea simply seems to be you can arbitrarily cheat, then enforce a tit-for-tat strategy until your opponent decides to give you another chance. As the modern "evolutionary" play styles seem to be built around cooperation and avoiding falling into long negative spirals, you gain an advantage. Certainly realistic, as I (as have we all) have seen these behaviors in the real world. Also not super surprising.
What I thought was interesting, and perhaps more important, was they seem to show that the player with the shortest memory controlled the game - that having a thousand turn memory didn't help against tit-for-tat, because you would end up playing tit-for-tat regardless of your larger strategy. This is an idea that I think should be explored further.
Overall it seems interesting but I imagine the applicability of the IPD to biology is somewhat limited, in that it doesn't compare the overall gains of the prisoners as a system to other prisoner's systems. i.e. a "winning" strategy very well may end up with a disproportionately large piece of a very small pie.
If extortion reigns, what drives these and other acts of selflessness?
The simple answer is ensuring survival of the species. Even "selfish" understand that it is better to live another day than not live at all.
I don't know about Spherical Cows , but I thought The Prisoners' Dilemma was that if he tried to escape he would get run down by a large rubbery sphere.
If I tell you a figure with three sides is a triangle, would you reply that since "triangle" is defined as a figure with three sides, I am really saying "a figure with three sides is a figure with three sides", and therefore I am not saying anything?
It is not that difficult. Being selfish is always the best for the individual, and short term sometimes entire species. The biggest lifeform on the planet is the Oregon Armillaria ostoyae, which unlike much of its fungus cousins kills its hosts. It has a huge advantage against not only other fungi but pretty much all other life. Long term when the forest it feeds off dies it will not fair nearly as well, that is why fungus as a whole is so benefaction, because a species that causes its own huge natural disasters cannot compete with one that strengthens and enriches its home.
Troll is not a replacement for I disagree.
Some species of birds and social insects routinely help raise another's brood. Even bacteria can cooperate, sticking to each other so that some may survive poison. If extortion reigns, what drives these and other acts of selflessness?
Lacking a human ego, the species in question naturally accept that they are all one and part of a larger whole. Therefore self sacrifice is innate because it leads to the survival of the whole.
Dyson is one of my science heros. cf. https://en.wikipedia.org/wiki/...
He is a notable subversive and a joker. He was once commissioned to write a paper for the US DOD regarding the use of nukes in Vietnam. He is pointing something else out in this paper (already three years old - not news), and it appears the irony is being missed.
This comment was written with the intention to opt out of advertising.
You're born gay or you're born straight
That is an overly simplistic and wrong view of human sexuality. Sure there are people who claim to know 100% whether they are gay or straight from their earliest memory. There are also plenty of people who recognize that to some degree they did choose to be gay, straight, or some other orientation. Things like your environment, your thought processes (especially masturbatory thought processes), and the cultural norms you live in can influence your sexual attraction and behavior.
Traditionally, 'most fit' is measured in the units of fecundity, the expected number of offspring that reach reproductive age. So, 'survival of the fittest' actually means 'survival of the ones that get most offspring'. Although it has been argued, by Karl Popper no less, that this is a tautology in its own right, some relatively recent research has shown that in pre-biotic life, this is not necessarily true. You might get something best characterized as 'survival of the first', i.e., if you're first in the game, you can out-wit better adapted individuals by sheer numbers. Although a new breed can produce more offspring, once this game is iterated, the advantage disappears after a few generations, and the status-quo is maintained.
So, essentially, nothing really trivial or tautological about 'survival of the fittest'. And going back to the article itself, it might be that the paper actually shows some circumstance in which survival of the fittest is untrue.
chance. next.
Arrange an Axelrod tournament with Dyson's AI competing with other AIs to see if it's able to rise to the top, or at least be able to beat Tit for Tat. So far the only demonstrated way of defeating Tit for Tat is an even more cooperative strategy of AIs working in tandem.
It takes lot of energy to wage constant wars with your peers, the energy which could be used to breed more off-spring and fighting and searching for a bigger territory which leads to more energy input for the community. It is more profitable to evolve along your pray than solely for the constant competition.
The existence of mating seasons prove that the evolution does produce the purely selfish and destructive behaviours, and various ape species do fuck up and deceive the weaker members of their community regularly, given the right circumstances. The point is that this does not happen constantly.
In real life, which so many experts seem uninterested in, the participants in the prisoner's dilemma need to keep quiet until they've talked to a lawyer.
Has anyone thought that there are multiple solutions to the question that are both effective....and THAT is why we see both in nature? Smh
i meant for this to be funny but it's probably more true than funny
Remember kids, if you're not paying for the service, YOU ARE THE PRODUCT THAT IS BEING SOLD.
Very insightful: https://www.quantamagazine.org...
"Carmi Turchick says: February 13, 2015 at 9:30 am
Agree with Ratcliff's last statement. The issue is considerably more complicated in humans than in bacteria, and even in bacteria one needs to consider how hostile the environment is. What is astonishing about most of the PD literature is how it claims to examine evolution but never mentions the environment. A hostile environment, as Dugatkin showed, selects for more cooperation. The free-living bacteria that under drought convictions form a colony that creates a stalk and spores are an example and they point to the next error, which is assuming a reward is always available no matter the actions of the players. This is not how nature works. If too few of the bacteria cooperate, no stalk is made, no spores are released, all of the bacteria have a fitness of zero. Similarly in humans there are many times when obtaining any reward requires N number of individuals to cooperate, and often that number is unknowable. Nine of us might kill that elephant, or it might be one or two or three too few to get it done resulting in nothing for all of us. Even with two partners, if you selfishly fail to cut off the monkey's escape route he gets away and we both go hungry. Think I will go hunting with you again? Which brings up yet another issue; avoiding detection and the cost of being detected. PD assumes that the cost of defecting is limited to a partner picking defect in the next round. Some models allow partners to punish a player at a significant cost to themselves or to move to another partner, but even these fall well short of what we see in human groups. As described by Boehm in "Heirarchy in the Forest," those whose selfish behavior is detected face collective punishment by the group, costing each group member very little, which ranges from social shunning to being murdered by one's own family or abandoned and left alone by the group. The power in a group of cooperators belongs to the cooperators and not the defectors, as cooperators work together to thwart defectors but defectors by definition cannot gang up on cooperators in return. As PD examines interactions with two parties, if the cooperator is paired with a defector or extorter they have no one to cooperate with. But in a group they have plenty of cooperative partners while the selfish stand alone. This imbalance of power means that the opportunities to defect are extremely limited as one must avoid detection, a situation which favors cooperation as the dominant and more numerous strategy. Finally, in group social territorial species having and defending a territory is an all or nothing issue with N number required to keep neighbors from taking your land and killing everyone. Either all of you have land and lives or none of you have land and at the very least few men and children survive. So we see that fairly often the "reward" for defecting is actually not 3 or whatever number is randomly chosen, but instead it is nothing, or loss of social status, or it is death for the individual, or death for the individual and all their relatives."
A 21st century issue: the irony of technologies of abundance in the hands of those still thinking in terms of scarcity.
Seriously? 2015 and supposed "scientists" are still treating evolution as if it happens at the species level? Selection happens at the gene level. And if you can't understand the implications of that, maybe you should go back to 1976 and read "The Selfish Gene". I guess Dawkins has wasted so much of his time in pointless arguments with US creationist loonies that people have forgotten that he solved the "mystery" of how cooperation arises from selfishness almost 40 years ago (and it was never really a mystery for geneticists or biologists with a clue, anyway, at least since the discovery of DNA).
misunderstanding Rand. Was it unintentonal (by not reading, or by having poor reading comprehension) or intentional?
Rand was NOT opposed to mercy or charity or goodwill, or sharing, etc; She was against the state or, or the masses, FORCING people to be faux-charitable (in the form of rolling over and playing dead while their pockets were picked by politicians and bureaucrats who then give the money to their favored beneficiaries). Rand wrote repeatedly that people have the absolute right to be as charitable and generous as THEY wish to be, but NOBODY has the moral right to force them to give their stuff away, and they certainly have no obligation to assist in their own abuse. These are two entirely different things.
Since you are on Slashdot, I'll assume you are not an idiot who lacks reading comprehension skills. Given your angry and dishonest anti-Rand rant, that means you are probably a Marxist; Marxism depends on the insistence that armed robbery, when comitted by government/masses, is actually a high form of charity. People with such a wordview, always detest Rand because her very strongly and carefully reasoned arguments undercut the basic mechanisms that Marxist utopians NEED.
This is like this whole lame-brained "nature vs. nurture" debate that raged for decades before everyone figured out "duh it's both, stupids".
The prisoner's dilemma & other ultra-oversimplified peeks at "life" determine which strategy or strategies will work best depending on both their rules, and in this case, what weighting factors (scores) are assigned to each outcome. You could easily tweak the payout matrix to "prove" other approaches are best, but what does that say about broader reality?
Nature having more than one niche, I'd figure you'd expect to find situations where co-operation works best, where competition works best, and every other conceivable strategy. Which is what we see. Thus leading to the obvious conclusion "the prisoner's dilemma is far simpler and far more uniform than life, and thus requires far fewer and simpler solutions". Duh, stupids.
From the tube-worm and archaea populated super-hot ocean floor water vents, to your living room, to the wilds of Africa, to the antarctic ice-floes (hi there, adorable waddling penguins!), reality calls for a variety of strategies and even some gender-shifting fish. Does the prisoner's dilemma "prove" to us the viability of gender-switching strategies? No, too narrow a model to even address the question. Or to address MOST questions.
Regards,
-- Dr. Cat
There is So Much Fail in the article.
Prisoner's Dilemma does not and never has suggested cooperation: just the opposite - it always suggests defection!! That was the insight of Nash.
However, Prisoner's Dilemma is only a specifically weighted, 1-iteration, 2-player game. Not all games that happen to be 2-player games or single iteration or both are Prisoner's Dilemma!! In fact, most are NOT!. MADD, for example, is NOT Prisoner's Dilemma because the weights are different!
Further, cooperation is more universally the solution for iterated games which is what Tit-for-Tat is about. Prisoner's Dilemma is NOT an iterated game. Any game that models evolution, however, is iterated. But there are no proofs for Tit-for-Tat being optimally cooperative except by exhaustive empirical testing.
Jesus Christ, the sloppiness of this entire discussion is breathe taking and depressing. No wonder the United States is about to start a nuclear war with Russia - people can't even keep basic game theory straight!
It is not just the selfish that can use extortion. It goes like this:
"Be polite, because if you piss me off bad enough I will shoot you. I might go to jail for life, but you will be dead."
Seems to work here in Virginia... (and y'all wonder why southerners are so polite). 8-)
The math sounds interesting.
But all they have really shown, is that the calculations are very sensitive to initial conditions. That is the problem with the calculations for predicting the weather, and you know how good that is... 8-)